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Theorem disjss2 4153
Description: If each element of a collection is contained in a disjoint collection, the original collection is also disjoint. (Contributed by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
disjss2  |-  ( A. x  e.  A  B  C_  C  ->  (Disj  x  e.  A C  -> Disj  x  e.  A B ) )

Proof of Theorem disjss2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ssel 3310 . . . . 5  |-  ( B 
C_  C  ->  (
y  e.  B  -> 
y  e.  C ) )
21ralimi 2749 . . . 4  |-  ( A. x  e.  A  B  C_  C  ->  A. x  e.  A  ( y  e.  B  ->  y  e.  C ) )
3 rmoim 3101 . . . 4  |-  ( A. x  e.  A  (
y  e.  B  -> 
y  e.  C )  ->  ( E* x  e.  A y  e.  C  ->  E* x  e.  A
y  e.  B ) )
42, 3syl 16 . . 3  |-  ( A. x  e.  A  B  C_  C  ->  ( E* x  e.  A y  e.  C  ->  E* x  e.  A y  e.  B
) )
54alimdv 1628 . 2  |-  ( A. x  e.  A  B  C_  C  ->  ( A. y E* x  e.  A
y  e.  C  ->  A. y E* x  e.  A y  e.  B
) )
6 df-disj 4151 . 2  |-  (Disj  x  e.  A C  <->  A. y E* x  e.  A
y  e.  C )
7 df-disj 4151 . 2  |-  (Disj  x  e.  A B  <->  A. y E* x  e.  A
y  e.  B )
85, 6, 73imtr4g 262 1  |-  ( A. x  e.  A  B  C_  C  ->  (Disj  x  e.  A C  -> Disj  x  e.  A B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1546    e. wcel 1721   A.wral 2674   E*wrmo 2677    C_ wss 3288  Disj wdisj 4150
This theorem is referenced by:  disjeq2  4154  0disj  4173  uniioombllem2  19436  uniioombllem4  19439  usgreghash2spotv  28177
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-ral 2679  df-rmo 2682  df-in 3295  df-ss 3302  df-disj 4151
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